Proposition 1 (Chordal sparse matrices [1]}) 
Let \( \mathcal {G}\)  be an undirected chordal graph with edges \(\mathcal {E}\) , vertices \(\mathcal {V}\)  and a set of maximal cliques \(\mathcal {T} = \lbrace C_1,\dots ,C_p \rbrace \) . The set of matrices with a sparsity pattern defined by \( \mathcal {G}\)  is defined as: \(\mathbb {S}^n(\mathcal {E} ) := \lbrace  X \in \mathbb {S}^n : X_{ij} = 0 \;\mathrm {if}\; (i,j) \notin \mathcal {E} \; \mathrm {for}\; i \ne j \rbrace  \) . Then \(X \in \mathbb {S}^n(\mathcal {E} ) \)  is positive semidefinite if and only if \(X = \sum ^p_{k=1} X_k ,\; X_k \ge 0, \; k \in [p]\) , where \(X_k \in S( C_k ) := \lbrace  X \in \mathbb {S}^n : X_{ij} = 0 \;\mathrm {if}\; (i,j) \notin C_k \times C_k \rbrace  \) .

